MIUI is a forked version of Android with plenty of additional features & tweaks not found in stock AOSP ROM. MIUI is very smooth & has a great battery life on the Redmi Note 3. The powerful Snapdragon 650 with its excellent single core performance coupled with dual channel LPDDR3 RAM & eMMC 5.1 flash storage results in blazing fast 1st time app opening speeds.

MIUI by default is very aggressive at killing background apps in the running in the memory, as a result it is unable to keep more than 3-4 apps in the memory while multi-tasking or after waking up the device. Default settings are aimed to keep as much free RAM as possible thats where the problem lies. It kills the apps running in the background to keep most of its RAM free. Apps get killed & have to be reloaded again from the flash storage into the memory, this takes a lot of CPU, I/O cycles & makes the end user experience less enjoyable. Below settings might make your multitasking a bit better at the cost of less free RAM, but that shouldn't matter as RAM is meant to be used not kept free. Always remember "Free RAM is Wasted RAM.

Lock Apps In Memory
- You can lock your desired / most frequently used apps in the memory.
- Open your app, let it get fully launched, open the recent apps menu.
- Swipe down the app icon or recent app screen to reveal a lock icon on top left corner.
- Tap on it, this locks the app into the memory & it wont get force killed in the background.
- To unlock the app repeat the same process and tap it again to unlock it from memory.

Important
- After doing the above mentioned changes, restart your device.
- You might have to re-apply these changes after every MIUI OTA update.
- These settings have hardly any effect on the battery life.
- I am getting more or less the same battery backup after using these settings.

Wiki : What does each and every option do?• Memory Optimization : This option sets the swap file for multi-tasking. Size varies according to the device and RAM configuration.
- High: 767MB to 1GB
- Middle: 767MB to 1GB
- Low: Swap file disabled
- Off: Swap file disabled• Show All ANRs : This option enables MIUI to show an obtrusive dialogue box after an app fails to stop responding . Setting this off doesnt show the pop-up frequently.
• Force-Closed Apps : This option enables MIUI to show an obtrusive dialogue box after an app force-closes unexpectedly. Setting this off doesnt show the message. It enables after every reboot.• Turn On MIUI Optimization : This option enables a plethora of MIUI based settings & optimizations as per the guidelines set by the MIUI developers. It helps to load the app and the app data in parallel reducing load times and ensuring snappier user experience. This setting can sometimes cause a variety of issues on non-MIUI based apps such as Google Apps & Apps from Google Playstore. Its better to disable if you rely on Google Playstore for apps & services or use Global Stable or Global Beta MIUI ROMs